Question: Haruka and Mustafa were asked to find an explicit formula for the sequence $4\,,\,12\,,\,36\,,\,108,...$, where the first term should be $g(1)$. Haruka said the formula is $g(n)=4\cdot3^{{n}}$, and Mustafa said the formula is $g(n)=4\cdot4^{{n-1}}$. Which one of them is right? Choose 1 answer: Choose 1 answer: (Choice A) A Only Haruka (Choice B) B Only Mustafa (Choice C) C Both Haruka and Mustafa (Choice D) D Neither Haruka nor Mustafa
Explanation: In a geometric sequence, the ratio between successive terms is constant. This means that we can move from any term to the next one by multiplying by a constant value. Let's calculate this ratio over the first few terms: $\dfrac{108}{36}=\dfrac{36}{12}=\dfrac{12}{4}={3}$ We see that the constant ratio between successive terms is ${3}$. In other words, we can find any term by starting with the first term and multiplying by ${3}$ repeatedly until we get to the desired term. Let's look at the first few terms expressed as products: $n$ $1$ $2$ $3$ $4$ $g(n)$ ${4}\cdot\!{3}^{0}$ ${4}\cdot\!{3}^{1}$ ${4}\cdot\!{3}^{2}$ ${4}\cdot\!{3}^{3}$ We can see that every term is the product of the first term, ${4}$, and a power of the constant ratio, ${3}$. Note that this power is always one less than the term number $n$. This is because the first term is the product of itself and plainly $1$, which is like taking the constant ratio to the zeroth power. Thus, we arrive at the following explicit formula (Note that ${4}$ is the first term and ${3}$ is the constant ratio): $g(n)={4}\cdot{3}^{{\,n-1}}$ We can now see that $g(n)=4\cdot3^{{\,n}}$ is not a correct formula, because the constant ratio is multiplied one extra time for each term. For instance, according to this formula, the value of the first term would be: $g(1)=4\cdot3^{{\,1}} = 12$. However, according to our table of values, $g(1)=4$. So Haruka is definitely wrong. What about Mustafa? We can see that $g(n)=4\cdot4^{{\,n-1}}$ is also not a correct formula, because the constant ratio according to this formula is $4$, while the actual constant ratio is $3$. Hence, Mustafa is also wrong. Neither Haruka nor Mustafa got a correct explicit formula.